Answers: (more advanced material is in blue; I am assuming any key-card discussion is more advanced)
1.
Hint: 5NT asks for kings. Is there any way out?
Answer: You were hoping that partner had at least three aces for his raise to 3D. Four aces would give you 13 top tricks in 7NT; 3 aces would make a diamond slam (
6NT is probably okay but why risk going down on a club lead if you're missing an ace and the ♣K or the ♣AQ? East can double 6NT for a club lead in that situation. If you're curious, look up Lightner Double.)
Sadly, partner only has two aces so you can't make any slam, 5NT asks for kings, and you are past 5D. Fortunately, there is a way to get out at the five level. Make an impossible bid, in this case 5S. You can't possibly want to play in 5S on this auction. What 5S says (and this is standard bridge) is "You gave me an embarrassing answer and I would like to get out in 5NT." Your partner should bid 5NT which you will gratefully pass. You should lose only the two aces, since the only way partner can have a 3D bid is to have great clubs. It is possible that someone might lead from ace-jack in a major suit, and you'll have to lose an ace in a minor suit to East who will lead the major back, but that would be very unlucky. However, staying at the five level is the only hope for a reasonable contract missing two aces, barring the unlikely event of partner having a void. I certainly would not bid 6D hoping for a void or a defensive mistake.
The recommended call is 5S.
Some of you play Roman Keycard Blackwood where the next step asks about the queen of trump. You may be asking why the 5S bid requests partner to bid 5NT and does not ask about the trump queen. The answer is that your 5H bid already denied the trump queen so there is no sense in asking for it; the next step (that can't be a final contract) only asks for the trump queen over a 5C or 5D response. If you play standard Blackwood, you should ignore this entire paragraph.
2.
Hint: What does partner have? Is it possible that you can go down in a slam?
Answer: Wow - you have 22 HCP (23 points counting length) and your partner showed 16-18 points. Sounds like you have a grand slam - but let's check and make sure we're not missing an ace.
Surprise! Partner only shows two aces, so the opponents do have an ace. We thought we had about 39 points but the opponents have an ace. Surely we must have everything else. Before bidding 6S, let's ask ourselves what could possibly go wrong if we are only missing an ace?
Hmm - somebody might get a ruff. Perhaps West has a singleton and East has either that ace or the trump ace. It's not likely but why take the chance? I suppose there is also the off-chance that someone has all the missing trumps and you can't avoid a trump loser and the ace.
When we think of what could go wrong, the solution is just around the corner. Partner has virtually all the missing high cards except the missing ace so 6NT should be easy - just give up the ace and take 12 tricks.
What does partner have is a more advanced question involving visualizing partner's hand. They showed 16 points but only with you having 22 HCP The opponents have an ace so partner has at most 14 HCP and must get the rest of his 3S bid from shortness which is almost certainly a singleton diamond. (It's pretty hard for him to get 14 HCP with a singleton heart and an ace missing.) So partner has something like: S-Axxx H-KJx D-x C-AJxxx. As you can see, you won't make 6S if West has ♠JTxx but can make 6NT. Or partner could have S-AJxx H-KJx D-A C-JTxxx or S-JTxx, H-KJx, D-A, C-AJxxx where a club ruff will set you but again 6NT makes by promoting the suit missing the ace.
3.
Remember - if you play RKC 0314, the bidding diagram should have 5D instead of 5C. If you don't know what I'm talking about, ignore this.
Hint: What contract is partner interested in when he bids 5NT?
Answer: If your partner was only interested in a small slam, he would have bid 6C. Partner must be interested in a grand slam if you have enough kings. This means that partner has all the aces. Partner can't be worried about trump quality since partner plans on bidding a grand slam if you show the right number of kings.
Or the right king if you play specific kings. Again, if you haven't heard of specific kings, don't worry about it. I haven't played them either
Since partner is looking for a grand slam and you have no aces, he must have all of them. In fact, when your partner asks for kings, he is promising that your side has all the aces and giving you permission to bid a grand slam if you have an undisclosed source of tricks that would make up for a missing king; i.e. you think there are thirteen tricks. Your partner has the SA, the HA, the DA giving you six diamond tricks, and solid clubs missing the king. If partner has five clubs or a king, you can count thirteen tricks in notrump, and 7NT is a reasonable bid, which will need a heart finesse if partner has only four clubs and no king.
However, 7C might be the right contract. Partner knows he is giving you permission to bid a grand slam with a source of tricks, and partner may be asking for kings with something like S-Axx H-Axx D-Axx C-AQJx hoping that you will bid the grand slam with solid diamonds, and settling for 6NT otherwise. You will need a spade ruff for your thirteenth trick to avoid a heart finesse.
I would probably choose 7NT since partner might have C-AQxxx and a side king where you have a shot to make 7NT but a club loser will doom 7C.
Take full credit for bidding 7NT or 7C. Take partial credit for planning to bid either of these after partner bids 6NT after your king answer. You only get partial credit because if partner thinks for a while and then bids 6NT, the director might rob you of a successful conversion to a grand slam.
Asking for kings gives permission to bid a grand slam. This means that you don't ask for kings when you have three aces and a void and partner doesn't show the other ace. It also means you don't ask for kings just to find out if 6NT is better than a suit slam. Partner's option to bid a grand slam with a source of tricks is too valuable to give up.